Educational Articles

A Primer on Put-Call Parity and How to Use It

Lawrence D. Cananagh
| October 06, 2009

This week, we review what are known as the put/call parity rules. If you know one rule - and you remember your high school algebra - you can quickly master all the rules. Mastery of these rules gives you a lot more flexibility when planning your option strategies.

Are Puts Cheaper than Calls?

(More than a hint: They’re not.)

Look at most option prices in which the stock price is close to the strike price. You are likely to see that put premiums are lower than call premiums. Are puts cheaper than calls? In fact, the time premiums of puts and calls at the same strike price (and the same expiration) are theoretically (and for practical purposes) the same. Why then do call premiums usually appear to be higher?

The answer is that with the stock equal to the strike, the calls are often in-the-money and the puts are out-of-the-money. This is because the real price of the underlying is the stock’s future delivery price, which is determined by the stock’s dividend and the going interest rate as well as by the stock price. If the interest rate is higher than the dividend rate (as it usually is) then the stock’s future delivery price will be higher than its current price.

Here is an example. If you are a market maker and you need to buy the stock (and lock in a price) for one-year delivery (for a call you are writing), you need to borrow the funds (at the one-year interest rate) to buy the stock. This increases your effective price of the stock. However, you also get to collect the dividends (if any) on the stock. This will reduce your price. Thus, if the current stock price is $100, the one-year interest rate is 6% and the dividend rate is 1% p.a., then the real cost of the stock for a one-year option is $105.

In Figure 1 below, we priced one-year calls and puts using a standard Black-Scholes model with the stock at $100, the interest rate at 6% and the dividend rate at 1%. At the $100 strike price, the call premium is higher than the put premium. At the $105 strike price (equal to the future delivery price), the call and put premiums are exactly equal. You will also notice that when you use the $105 future delivery price as your underlying, rather than the $100 current stock price, all the calculated time premiums of puts and calls for each strike price are equal.

An Arbitrage Opportunity

“Fine,” you might say “that’s the theory, but is it the way options really trade?”

The answer is “yes.” The reason is that when call or put time premiums get out of line with each other, option market makers can make a risk-free arbitrage profit.

Let us look at the simplest arbitrage. Suppose that the time premium of a particular put is greater than the time premium of the corresponding call (same stock, strike and expiration). Here the market maker can take advantage of the difference in the time premiums by writing the put and buying the call. What the market maker has done is to create a “synthetic” long stock position with a free credit of premium. The market maker can fully offset his risk by selling the stock. The net credit of time premium from this transaction will be his arbitrage profit.

From the above, you get the first put/ call parity rule: If you buy a call and sell a put at the same strike price and expiration, you get the equivalent of a long stock position. Or,

Rule 1: (+) Buy Call & (-) Write Put = (+) Buy Stock

From the aforementioned, it follows that if you write a call and buy a put at the same strike price, you get the equivalent of a short stock position.

Rule 2: (-) Write Call & (+) Buy Put = (-) Short Stock

The other put call parity rules come from the fact that option time premiums are essentially the same for calls and puts at the same strike price.

Rule 3: (+) Buy Stock & (+) Buy Put = (+) Buy Call

Rule 4: (-) Short Stock & (-) Write Put = (-) Write Call

Rule 5: (-) Short Stock & (+) Buy Call = (+) Buy Put

Rule 6: (+) Buy Stock & (-) Write Call = (+) Write Put = Covered Call

Notice that if you move any of these transactions to the other side of the equal sign, you change the strategy from a short to a long (or vice versa) and (remembering your algebra) the “+” sign to a “-” sign (or vice versa).

Uses of Put/Call Parity

Can I take advantage of the put/call parity rules? Because the market maker has a clear advantage in trading options, it is unlikely that the typical investor will see actual “risk-free” arbitrage opportunities from put/call parity. However, these rules are useful, because knowing them can help you quickly determine what your strategy alternatives are. Here are some applications

Rule 3: Often, you may be interested in buying a particular stock, but you are concerned about the risk and may want to hedge it with a put. Knowing rule number 3 tells you that buying the call will give you the same risk/reward as buying the stock and the put.

Rule 6: Often, you may see attractive covered call opportunities, where the call is in-the-money. Here you may be reluctant to establish the position in fear of having the call exercised. Here you can write a cash covered put instead of the covered call. In Figure 2 on page 3, we show a comparison between an in-the-money covered call (the November $22.50 call) and the corresponding out of-the-money put write (also November $22.50 strike) on Internet Security (ISSX) with the stock at $23.21. Although the cash covered put offers a somewhat lower per annum yield than the covered call (51.0% versus 57%), it offers investors the advantage of no closing transactions if the stock ends up above the $22.50 strike price. (Subscribers may want to review our report, “Cash Covered Puts versus Covered Calls,” Ot050509.Pdf, which compares these two strategies side by side.)